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#############################################################################
##
#W pcpelms.gi Polycyc Bettina Eick
##
InstallGlobalFunction( PcpElementConstruction,
function( coll, list, word )
local elm;
elm := rec( collector := coll,
exponents := Immutable(list),
word := Immutable(word),
name := "g" );
# objectify and return
return Objectify( coll![PC_PCP_ELEMENTS_TYPE], elm );
end );
#############################################################################
##
## Functions to create pcp elements by exponent vectors or words.
## In the NC versions we assume that elements are in normal form.
## In the other versions we collect before we return an element.
##
InstallGlobalFunction( PcpElementByExponentsNC,
function( coll, list )
local i, word;
word := ObjByExponents( coll, list );
return PcpElementConstruction( coll, list, word );
end );
InstallGlobalFunction( PcpElementByExponents, function( coll, list )
local h, k;
if Length(list) > NumberOfGenerators(coll) then
Error( "more exponents than generators" );
fi;
h := ObjByExponents( coll, list );
k := list * 0;
while CollectWordOrFail( coll, k, h ) = fail do od;
return PcpElementByExponentsNC( coll, k );
end );
InstallGlobalFunction( PcpElementByGenExpListNC,
function( coll, word )
local list, i;
list := ExponentsByObj( coll, word );
word := ObjByExponents( coll, list );
return PcpElementConstruction( coll, list, word );
end );
InstallGlobalFunction( PcpElementByGenExpList, function( coll, word )
local k;
k := [1..coll![PC_NUMBER_OF_GENERATORS]] * 0;
while CollectWordOrFail( coll, k, word ) = fail do od;
return PcpElementByExponentsNC( coll, k );
end );
#############################################################################
##
#A Basic attributes of pcp elements - for IsPcpElementRep
##
InstallMethod( Collector,
"for pcp groups",
[ IsPcpGroup ],
G -> Collector( One(G) ) );
InstallMethod( Collector,
"for pcp elements",
[ IsPcpElementRep ],
g -> g!.collector );
InstallMethod( Exponents,
"for pcp elements",
[ IsPcpElementRep ],
g -> g!.exponents );
InstallMethod( NameTag,
"for pcp elements",
[ IsPcpElementRep ],
g -> g!.name );
InstallMethod( GenExpList,
"for pcp elements",
[ IsPcpElementRep ],
g -> g!.word );
InstallMethod( Depth,
"for pcp elements",
[ IsPcpElementRep ],
function( elm )
if Length(elm!.word) = 0 then
return elm!.collector![PC_NUMBER_OF_GENERATORS] + 1;
else
return elm!.word[1];
fi;
end );
InstallMethod( TailOfElm,
"for pcp elements",
[ IsPcpElement and IsPcpElementRep ],
function( elm )
if Length( elm!.word ) = 0 then
return 0;
else
return elm!.word[ Length(elm!.word) - 1 ];
fi;
end );
InstallMethod( LeadingExponent,
"for pcp elements",
[ IsPcpElementRep ],
function( elm )
if Length(elm!.word) = 0 then
return fail;
else
return elm!.word[2];
fi;
end );
## Note, that inverses of generators with relative order > 0 are not treated
## as inverses as they should never appear here with a negative exponent.
BindGlobal( "IsGeneratorOrInverse", function( elm )
return Length(elm!.word) = 2 and
(elm!.word[2] = 1 or elm!.word[2] = -1);
end );
##
## Is elm the power of a generator modulo depth d?
## If so, then return the power, otherwise return fail;
##
BindGlobal( "IsPowerOfGenerator", function( elm, d )
if Length( elm!.word ) = 0 or
(Length( elm!.word ) > 2 and elm!.word[3] <= d) then
return fail;
fi;
return elm!.word[2];
end );
#############################################################################
##
#F FactorOrder( g )
##
InstallMethod( FactorOrder, [IsPcpElement],
function( g )
if Length( g!.word ) = 0 then return fail; fi;
return RelativeOrders( Collector(g) )[Depth(g)];
end );
#############################################################################
##
#F RelativeOrderPcp( g )
##
InstallMethod( RelativeOrderPcp, [IsPcpElement],
function( g )
local r, l;
if Length( g!.word ) = 0 then return fail; fi;
r := FactorOrder( g );
# the infinite case
if r = 0 then return 0; fi;
# the finite case
l := LeadingExponent( g );
if l = 1 then
return r;
elif IsBound( g!.normed ) and g!.normed then
return r / LeadingExponent(g);
elif IsPrime( r ) then
return r;
else
return r / Gcd( r, l );
fi;
end );
# TODO: Replace this by something like DeclareSynonymAttr.
# However, we cannot use DeclareSynonymAttr directly, because for
# collectors there is already an operation SetRelativeOrder.
BindGlobal( "RelativeOrder", function( g )
return RelativeOrderPcp(g);
end );
#############################################################################
##
#F RelativeIndex( g )
##
InstallMethod( RelativeIndex, [IsPcpElement],
function( g )
local r, l;
if Length( g!.word ) = 0 then return fail; fi;
r := FactorOrder( g );
l := LeadingExponent( g );
if IsBound( g!.normed ) and g!.normed then
return l;
elif r > 0 then
return Gcd( r, l );
else
return AbsInt( l );
fi;
end );
#############################################################################
##
#F Order( g )
##
InstallMethod( Order, [IsPcpElement],
function( g )
local o, r;
o := 1;
while g <> g^0 do
r := RelativeOrderPcp( g );
if r = 0 then return infinity; fi;
o := o*r;
g := g^r;
od;
return o;
end );
#############################################################################
##
#F NormingExponent( g ) . . . . . . . . .returns f such that g^f is normed
##
## Note that g is normed, if the LeadingExponent of g is its RelativeIndex.
##
BindGlobal( "NormingExponent", function( g )
local r, l, e;
r := FactorOrder( g );
l := LeadingExponent( g );
if IsBool( l ) then
return 1;
elif r = 0 and l < 0 then
return -1;
elif r = 0 then
return 1;
elif IsPrime( r ) then
return l^-1 mod r;
else
e := Gcdex( r, l ); # = RelativeIndex
return e.coeff2 mod r; # l * c2 = e mod r
fi;
end );
#############################################################################
##
#F NormedPcpElement( g )
##
BindGlobal( "NormedPcpElement", function( g )
local h;
h := g^NormingExponent( g );
h!.normed := true;
return h;
end );
#############################################################################
##
#M Print pcp elements
##
InstallMethod( PrintObj,
"for pcp elements",
[IsPcpElement],
function( elm )
local g, l, e, d;
g := NameTag( elm );
e := Exponents( elm );
d := Depth( elm );
if d > Length( e ) then
Print("id");
elif e[d] = 1 then
Print(Concatenation(g,String(d)));
else
Print(Concatenation(g,String(d)),"^",e[d]);
fi;
for l in [d+1..Length(e)] do
if e[l] = 1 then
Print("*",Concatenation(g,String(l)));
elif e[l] <> 0 then
Print("*",Concatenation(g,String(l)),"^",e[l]);
fi;
od;
end );
InstallMethod( String,
"for pcp elements",
[IsPcpElement],
function( elm )
local g, l, e, d, str;
g := NameTag( elm );
e := Exponents( elm );
d := Depth( elm );
if d > Length( e ) then
return "id";
fi;
str := Concatenation(g,String(d));
if e[d] <> 1 then
Append(str, Concatenation("^",String(e[d])));
fi;
for l in [d+1..Length(e)] do
if e[l] = 0 then continue; fi;
Append(str, Concatenation("*",g,String(l)));
if e[l] <> 1 then
Append(str, Concatenation("^",String(e[l])));
fi;
od;
return str;
end );
#############################################################################
##
#M g * h
##
InstallMethod( \*,
"for pcp elements",
IsIdenticalObj,
[IsPcpElement, IsPcpElement],
20,
function( g1, g2 )
local clt, e, f;
clt := Collector( g1 );
if TailOfElm( g1 ) < Depth( g2 ) then
e := Exponents( g1 ) + Exponents( g2 );
else
e := ShallowCopy( Exponents( g1 ) );
f := GenExpList( g2 );
while CollectWordOrFail( clt, e, f ) = fail do
e := ShallowCopy( Exponents( g1 ) );
od;
fi;
return PcpElementByExponentsNC( clt, e );
end );
#############################################################################
##
#M Inverse
##
InstallMethod( Inverse,
"for pcp elements",
[IsPcpElement],
function( g )
local clt, k;
clt := Collector( g );
if IsGeneratorOrInverse( g ) and RelativeOrderPcp(g) = 0 then
if LeadingExponent( g ) = 1 then
k := clt![PC_INVERSES][ Depth(g) ];
else
k := clt![PC_GENERATORS][ Depth(g) ];
fi;
else
k := FromTheLeftCollector_Inverse( clt, GenExpList(g) );
fi;
return PcpElementByGenExpListNC( clt, k );
end );
InstallMethod( InverseMutable,
"for pcp elements",
[IsPcpElement],
function( g )
local clt, k;
clt := Collector( g );
if IsGeneratorOrInverse( g ) and RelativeOrderPcp(g) = 0 then
if LeadingExponent( g ) = 1 then
k := clt![PC_INVERSES][ Depth(g) ];
else
k := clt![PC_GENERATORS][ Depth(g) ];
fi;
else
k := FromTheLeftCollector_Inverse( clt, GenExpList(g) );
fi;
return PcpElementByGenExpListNC( clt, k );
end );
#############################################################################
##
#M \^
##
InstallMethod( \^,
"for a pcp element and an integer",
[IsPcpElement, IsInt],
SUM_FLAGS + 10,
function( g, d )
local res;
# first catch the trivial cases
if d = 0 then
return PcpElementByExponentsNC( Collector(g), 0*Exponents(g) );
elif d = 1 then
return g;
elif d = -1 then
return Inverse(g);
fi;
# # use collector function
# c := Collector(g);
# k := FromTheLeftCollector_Power(c, ObjByExponents(c, Exponents(g)), d);
# return PcpElementByGenExpListNC( c, k );
# set up for computation
if d < 0 then
g := Inverse(g);
d := -d;
fi;
# compute power
res := g^0;
while d > 0 do
if d mod 2 = 1 then res := res * g; fi;
d := QuoInt( d, 2 );
if d <> 0 then g := g * g; fi;
od;
return res;
end );
InstallMethod( \^,
"for two pcp elements",
IsIdenticalObj,
[IsPcpElement, IsPcpElement],
function( h, g )
local clt, conj;
clt := Collector( g );
if IsGeneratorOrInverse( h ) and IsGeneratorOrInverse( g ) then
if Depth( g ) = Depth( h ) then
conj := h;
elif Depth( g ) < Depth( h ) then
conj := GetConjugateNC( clt,
Depth( h ) * LeadingExponent( h ),
Depth( g ) * LeadingExponent( g ) );
conj := PcpElementByGenExpListNC( clt, conj );
elif Depth( g ) > Depth( h ) then
# h^g = g^-1 * h * g
conj := ShallowCopy( Exponents( g^-1 ) );
while CollectWordOrFail( clt, conj,
[ Depth(h), LeadingExponent( h ),
Depth(g), LeadingExponent( g ) ] ) = fail do
conj := ShallowCopy( Exponents( g^-1 ) );
od;
conj := PcpElementByExponentsNC( clt, conj );
fi;
elif Depth(g) = TailOfElm(g) and Depth( g ) < Depth( h ) then
##
## nicht klar ob dies etwas bringt
##
g := [ Depth(g), LeadingExponent(g) ];
conj := ShallowCopy( Exponents( h ) );
while CollectWordOrFail( clt, conj, g ) = fail do
conj := ShallowCopy( Exponents( h ) );
od;
conj[ g[1] ] := 0;
conj := PcpElementByExponentsNC( clt, conj );
else
conj := g^-1 * h * g;
fi;
return conj;
end );
InstallMethod( GetCommutatorNC,
"for from the left collector",
[ IsFromTheLeftCollectorRep, IsInt, IsInt ],
function( coll, h, g )
if g > 0 then
if h > 0 then
if IsBound( coll![PC_COMMUTATORS][h] ) and
IsBound( coll![PC_COMMUTATORS][h][g] ) then
return coll![PC_COMMUTATORS][h][g];
else
return fail;
fi;
else
h := -h;
if IsBound( coll![PC_INVERSECOMMUTATORS][h] ) and
IsBound( coll![PC_INVERSECOMMUTATORS][h][g] ) then
return coll![PC_INVERSECOMMUTATORS][h][g];
else
return fail;
fi;
fi;
else
g := -g;
if h > 0 then
if IsBound( coll![PC_COMMUTATORSINVERSE][h] ) and
IsBound( coll![PC_COMMUTATORSINVERSE][h][g] ) then
return coll![PC_COMMUTATORSINVERSE][h][g];
else
return fail;
fi;
else
h := -h;
if IsBound( coll![PC_INVERSECOMMUTATORSINVERSE][h] ) and
IsBound( coll![PC_INVERSECOMMUTATORSINVERSE][h][g] ) then
return coll![PC_INVERSECOMMUTATORSINVERSE][h][g];
else
return fail;
fi;
fi;
fi;
end );
#############################################################################
##
#M Comm
##
InstallMethod( Comm,
"for two pcp elements",
[ IsPcpElement, IsPcpElement ],
function( h, g )
local clt, conj, ev;
clt := Collector( g );
if IsGeneratorOrInverse( h ) and IsGeneratorOrInverse( g ) then
if Depth( g ) = Depth( h ) then return g^0; fi;
if Depth( g ) < Depth( h ) then
## Do we know the commutator?
conj := GetCommutatorNC( clt, Depth( h ) * LeadingExponent( h ),
Depth( g ) * LeadingExponent( g ) );
if conj <> fail then
return conj;
fi;
## [h,g] = h^-1 h^g
conj := GetConjugateNC( clt, Depth( h ) * LeadingExponent( h ),
Depth( g ) * LeadingExponent( g ) );
ev := ShallowCopy( Exponents( h^-1 ) );
while CollectWordOrFail( clt, ev, conj ) = fail do
ev := ShallowCopy( Exponents( h^-1 ) );
od;
return PcpElementByExponentsNC( clt, ev );
fi;
if Depth( g ) > Depth( h ) and RelativeOrderPcp( g ) = 0 then
## [h,g] = (g^-1)^h * g
conj := GetConjugateNC( clt, Depth( g ) * -LeadingExponent( g ),
Depth( h ) * LeadingExponent( h ) );
ev := ExponentsByObj( clt, conj );
while CollectWordOrFail( clt, ev, GenExpList(g) ) = fail do
ev := ExponentsByObj( clt, conj );
od;
return PcpElementByExponentsNC( clt, ev );
fi;
fi;
return PcpElementByGenExpListNC( clt,
FromTheLeftCollector_Solution( clt,
GenExpList(g*h),GenExpList(h*g) ) );
end );
#############################################################################
##
#M One
##
InstallMethod( One, "for pcp elements", [IsPcpElement],
g -> PcpElementByExponentsNC( Collector(g), 0*Exponents(g) ) );
#############################################################################
##
#M \=
##
InstallMethod( \=,
"for pcp elements",
IsIdenticalObj,
[IsPcpElement, IsPcpElement],
function( g, h )
return Exponents( g ) = Exponents( h );
end );
#############################################################################
##
#M \<
##
InstallMethod( \<,
"for pcp elements",
IsIdenticalObj,
[IsPcpElement, IsPcpElement],
function( g, h )
return Exponents( g ) > Exponents( h );
end );
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